thats math

wat?

Kid. You won't what they got. Seriously, when I was young, I'm a smartass.

I needed 10 minutes to dig out my old textbook for the formulae for the coefficients and 3 minutes to do the question, and Hoppo did it in seconds.

In her head.

With no references.

Hoppo is the Kancolle version of Anzu, minus the laziness.

So, after trying to type out mathematical formulae with HTML code in the notes section (to provide a step-by-step explanation and calculation steps of how Hoppo got her answer), I gave up. Can't get the symbols to line up properly.

This is the reference I used. Anybody know how to make it work?

Edit: Saw the translation_request tag. Hmm, does this count as the first 'translation' here for math?

NNescio said:

Edit: Saw the translation_request tag. Hmm, does this count as the first 'translation' here for math?

Haha, sorry about that. When I put this tag, I was more thinking about the Admiral sign than the math (which give me a headach seriously lol)

Danbooru's safe list doesn't include MathML tags ?

Alright, some explanation for the Fourier series. I'm assuming that you have knowledge of at least AP Calculus or A-Level Pure Maths, or an equivalent. I could point you to some resources if you don't, but well, it's going to take some time to understand, and I don't think you'd be interested in the following explanation anyway.

'kay, I'll go into periodic functions first. Basically a periodic function is a function ("graph" in layman speak, but this is not quite correct) which repeats its values over regular intervals. The most (in)famous examples are probably the trigonometric functions, like sin(x), cos(x), tan(x), etc.

Now, what the Fourier series does it that it allows us to represent most periodic functions—even complex ones—as a sum of sines and cosines, which are generally far simpler. Sound waves are also periodic. A guitar chord, for example, will have a complicated wave pattern. Fourier analysis allows us to separate the complicated wave pattern into its component notes (represented by the sine and cosine functions) and the intensity of each note (represented by the coefficients of the respective sine and cosine functions).

In order to have a Fourier series representation, a periodic function must be piecewise differentiable (in its period). Now if you remember from your Calculus class, there are several criteria you need to prove before you can call a function differentiable, and they involve left and right-handed limits. I suspect many of you hated that.

We are not going to care about those, since this isn't a rigorous math course. Let's go with something easy—if you can draw a tangent line through every point of the graph for a function, then it's differentiable at every point. Naturally, if there are any sharp bends in the curve, like in |x|, then the function is not differentiable there, because we can't 'agree' on what the tangent line is at that point (it 'can' either be like its immediate left neighbor or is right, but we can't find out no matter how much we zoom in). Likewise if there's any discontinuity (like say, at the asymptote for 1/x) then the tangent line doesn't exist at that point, because, well, that point doesn't even exist.

Now, with piecewise differentiability, we're not even going to let those silly things get in our way. If we can 'chop' a function (or the graph of a function) into intervals ('pieces'), and each piece is differentiable at every point, then it's piecewise differentiable. We don't care about whether the 'pieces' are differentiable at their ends (the 'points' between each piece).

Using this definition |x| and 1/x are piecewise differentiable because we can 'cut' the 'graph' around 0 into two separate pieces, and each piece is differentiable (except at the ends, which we don't care).

Even with the relaxed criteria, a lot of functions are still not piecewise differentiable. A classic example is sin(1/x). The function oscillates faster and faster between -1 and 1 the closer we get to 0 (try zooming in and see). No matter where we cut the graph around 0, or how many times we cut, we cannot isolate the curve into nice, differentiable pieces. In effect, sin(1/x) has an infinite number of non-differentiable points in any interval centered around 0 (even if the interval is finite).

Alright, with all that said (phew!) we can finally move into the Fourier series:

For any periodic function f(x) with period p = 2L that is piece-wise differentiable in the interval [-L,L], there exists a Fourier series representation of f, given by:

F(x) = a_0 + ∑{n equals 1 to infinity}[a_n*cos(nπx) + b_n*sin(nπx)]

(Remember, n is the index for summation, and it starts from 1, so it must be an integer. This is important later.)

Where a_0, a_n, and b_n are coefficients given by the Euler-Fourier formulae (won't bore you with the proof, this is long enough):

a_0 = (1/2L)*∫{from -L to L} f(x) dx
a_n = (1/L)*∫{from -L to L} f(x)*cos(nπx/L) dx
a_n = (1/L)*∫{from -L to L} f(x)*sin(nπx/L) dx

Now going back to Ashigara's question, we have f(x)={0 if -1 ≤ x < 0; 1 if 0 ≤ x < 1. This function is periodic and piecewise differentiable (just 'chop' around 0). The period of the function is from -1 to 1, or 2, so p = 2L = 2, L = 1. Then:

a_0 = (1/2)*∫{from -1 to 1} f(x) dx
= (1/2)*[∫{from -1 to 0} f(x) dx + ∫{from 0 to 1} f(x) dx]
= (1/2)*[∫{from -1 to 0} 0 dx + ∫{from 0 to 1} 1 dx]
= (1/2)*[0 + (1 - 0)]
= 1/2

a_n = (1/1)*∫{from -1 to 1} f(x)*cos(nπx/1) dx
= ∫{from -1 to 0} f(x)*cos(nπx) dx + ∫{from 0 to 1} f(x)*cos(nπx) dx
= ∫{from -1 to 0} 0*cos(nπx) dx + ∫{from 0 to 1} 1*cos(nπx) dx
= 0 + ∫{from 0 to 1} (1/nπ)*nπ*cos(nπx) dx {Alternatively, use the substitution method
= (1/nπ)*[sin(nπ)-sin(0)]
= (1/nπ)*[0 - 0] {sine of any integer multiple of π is zero.
= 0

b_n = (1/1)*∫{from -1 to 1} f(x)*sin(nπx/1) dx
= ∫{from -1 to 0} f(x)*sin(nπx) dx + ∫{from 0 to 1} f(x)*sin(nπx) dx
= ∫{from -1 to 0} 0*sin(nπx) dx + ∫{from 0 to 1} 1*sin(nπx) dx
= 0 + ∫{from 0 to 1} (-1/nπ)*nπ*-sin(nπx) dx {Alternatively, use the substitution method
= (-1/nπ)*[cos(nπ)-cos(0)]
= (1/nπ)*[1 - cos(nπ)]
= (1/nπ)*[1 - (-1)^n] {cos(nπ) = -1 when n is odd, and 1 when n is even

Putting it all together:

F(x) = a_0 + ∑{n equals 1 to infinity}[a_n*cos(nπx) + b_n*sin(nπx)]
= 1/2 + ∑{n equals 1 to infinity}[0*cos(nπx) + (1/nπ)*[1 - (-1)^n]*sin(nπx)]
= 1/2 + ∑{n equals 1 to infinity}[(1/nπ)*(1 - (-1)^n)*sin(nπx)]

Exactly like what Hoppo said.

All in all, this is an elegant question, as there are a lot of parts which can be simplified, allowing us to do it quickly if we know the trick. No tedious calculations at all (the actual steps I used are far less and I did it mentally, but I had to write down the answer as I go, unlike Hoppo.) Notably we can ignore any of integrals from -1 to 0 since they are going to end up as zero anyway. Also, the calculation for a_n can be done very quickly if you realize you are going to end up with sin(nπ) on the right side of the equation after integrating cos(nπx), so you can skip all the middle steps and don't even bother with the constants.

Edit: Moved some negative signs around to make things slightly more legible.

NNescio said:

A ton of math and words

Holy ball sacks. Okay, first off you're amazing for all of that. I admiral your effort in writing an entire book here.

Buuuut....you know what, I don't know what to say. I'm speechless to know that someone actually went and tried to solve the math problem.

This is the first time Hoppo didn't screw up the scene.

Now i'm wondering how high Hoppo's IQ is..

Also NNescio, omega good job with the math.. I still don't get it as i'm an art student (where i hailed from, they said that art student is 'alergic' to math, half true for me) but i could see why people would regard math as an art.. It's almost, playful..

NNescio said:

Alright, some explanation for the Fourier series. I'm assuming that you have knowledge of at least AP Calculus or A-Level Pure Maths, or an equivalent. I could point you to some resources if you don't, but well, it's going to take some time to understand, and I don't think you'd be interested in the following explanation anyway.

'kay, I'll go into periodic functions first. Basically a periodic function is a function ("graph" in layman speak, but this is not quite correct) which repeats its values over regular intervals. The most (in)famous examples are probably the trigonometric functions, like sin(x), cos(x), tan(x), etc.

Now, what the Fourier series does it that it allows us to represent most periodic functions—even complex ones—as a sum of sines and cosines, which are generally far simpler. Sound waves are also periodic. A guitar chord, for example, will have a complicated wave pattern. Fourier analysis allows us to separate the complicated wave pattern into its component notes (represented by the sine and cosine functions) and the intensity of each note (represented by the coefficients of the respective sine and cosine functions).

In order to have a Fourier series representation, a periodic function must be piecewise differentiable (in its period). Now if you remember from your Calculus class, there are several criteria you need to prove before you can call a function differentiable, and they involve left and right-handed limits. I suspect many of you hated that.

We are not going to care about those, since this isn't a rigorous math course. Let's go with something easy—if you can draw a tangent line through every point of the graph for a function, then it's differentiable at every point. Naturally, if there are any sharp bends in the curve, like in |x|, then the function is not differentiable there, because we can't 'agree' on what the tangent line is at that point (it 'can' either be like its immediate left neighbor or is right, but we can't find out no matter how much we zoom in). Likewise if there's any discontinuity (like say, at the asymptote for 1/x) then the tangent line doesn't exist at that point, because, well, that point doesn't even exist.

Now, with piecewise differentiability, we're not even going to let those silly things get in our way. If we can 'chop' a function (or the graph of a function) into intervals ('pieces'), and each piece is differentiable at every point, then it's piecewise differentiable. We don't care about whether the 'pieces' are differentiable at their ends (the 'points' between each piece).

Using this definition |x| and 1/x are piecewise differentiable because we can 'cut' the 'graph' around 0 into two separate pieces, and each piece is differentiable (except at the ends, which we don't care).

Even with the relaxed criteria, a lot of functions are still not piecewise differentiable. A classic example is sin(1/x). The function oscillates faster and faster between -1 and 1 the closer we get to 0 (try zooming in and see). No matter where we cut the graph around 0, or how many times we cut, we cannot isolate the curve into nice, differentiable pieces. In effect, sin(1/x) has an infinite number of non-differentiable points in any interval centered around 0 (even if the interval is finite).

Alright, with all that said (phew!) we can finally move into the Fourier series:

For any periodic function f(x) with period p = 2L that is piece-wise differentiable in the interval [-L,L], there exists a Fourier series representation of f, given by:

F(x) = a_0 + ∑{n equals 1 to infinity}[a_n*cos(nπx) + b_n*sin(nπx)]

(Remember, n is the index for summation, and it starts from 1, so it must be an integer. This is important later.)

Where a_0, a_n, and b_n are coefficients given by the Euler-Fourier formulae (won't bore you with the proof, this is long enough):

a_0 = (1/2L)*∫{from -L to L} f(x) dx
a_n = (1/L)*∫{from -L to L} f(x)*cos(nπx/L) dx
a_n = (1/L)*∫{from -L to L} f(x)*sin(nπx/L) dx

Now going back to Ashigara's question, we have f(x)={0 if -1 ≤ x < 0; 1 if 0 ≤ x < 1. This function is periodic and piecewise differentiable (just 'chop' around 0). The period of the function is from -1 to 1, or 2, so p = 2L = 2, L = 1. Then:

a_0 = (1/2)*∫{from -1 to 1} f(x) dx
= (1/2)*[∫{from -1 to 0} f(x) dx + ∫{from 0 to 1} f(x) dx]
= (1/2)*[∫{from -1 to 0} 0 dx + ∫{from 0 to 1} 1 dx]
= (1/2)*[0 + (1 - 0)]
= 1/2

a_n = (1/1)*∫{from -1 to 1} f(x)*cos(nπx/1) dx
= ∫{from -1 to 0} f(x)*cos(nπx) dx + ∫{from 0 to 1} f(x)*cos(nπx) dx
= ∫{from -1 to 0} 0*cos(nπx) dx + ∫{from 0 to 1} 1*cos(nπx) dx
= 0 + ∫{from 0 to 1} (1/nπ)*nπ*cos(nπx) dx {Alternatively, use the substitution method
= (1/nπ)*[sin(nπ)-sin(0)]
= (1/nπ)*[0 - 0] {sine of any integer multiple of π is zero.
= 0

b_n = (1/1)*∫{from -1 to 1} f(x)*sin(nπx/1) dx
= ∫{from -1 to 0} f(x)*sin(nπx) dx + ∫{from 0 to 1} f(x)*sin(nπx) dx
= ∫{from -1 to 0} 0*sin(nπx) dx + ∫{from 0 to 1} 1*sin(nπx) dx
= 0 + ∫{from 0 to 1} (-1/nπ)*nπ*-sin(nπx) dx {Alternatively, use the substitution method
= (-1/nπ)*[cos(nπ)-cos(0)]
= (1/nπ)*[1 - cos(nπ)]
= (1/nπ)*[1 - (-1)^n] {cos(nπ) = -1 when n is odd, and 1 when n is even

Putting it all together:

F(x) = a_0 + ∑{n equals 1 to infinity}[a_n*cos(nπx) + b_n*sin(nπx)]
= 1/2 + ∑{n equals 1 to infinity}[0*cos(nπx) + (1/nπ)*[1 - (-1)^n]*sin(nπx)]
= 1/2 + ∑{n equals 1 to infinity}[(1/nπ)*(1 - (-1)^n)*sin(nπx)]

Exactly like what Hoppo said.

All in all, this is an elegant question, as there are a lot of parts which can be simplified, allowing us to do it quickly if we know the trick. No tedious calculations at all (the actual steps I used are far less and I did it mentally, but I had to write down the answer as I go, unlike Hoppo.) Notably we can ignore any of integrals from -1 to 0 since they are going to end up as zero anyway. Also, the calculation for a_n can be done very quickly if you realize you are going to end up with sin(nπ) on the right side of the equation after integrating cos(nπx), so you can skip all the middle steps and don't even bother with the constants.

Edit: Moved some negative signs around to make things slightly more legible.

What this confusing math do for ?
measure against something ?
or what ?
don't tell me all this math just to make sure road was straight...

NNescio said:

lots of Fourier analysis

I'm actually kind of jealous - my college calculus never actually went into detail on Fourier analysis. I remember being fascinated by the concept of the Fourier transform - that there was a two-way equivalence relation between the time domain and the frequency domain - but I never got to study how to actually do it.

LCerberus said:

What this confusing math do for ?
measure against something ?
or what ?
don't tell me all this math just to make sure road was straight...

The Fourier series leads to its generalized ('extended') counterpart, the Fourier Transform. The latter (or rather, a variant, called FTT, or Fast Fourier Transform) is responsible for why you can watch Youtube videos and surf for images on Danbooru. And reading this comment I posted, for that matter.

All current WiFi technology is dependent on the Fourier Transform (And by extension, the Fourier Series). Oh and your cellphone probably use Fourier Transforms too to communicate with cell towers set up by your mobile carrier.

Also CSI analysis of chemical residues, MRI scans in hospitals, some encryption algorithms, most compression algorithms (like your favorite MP3s if you happen to listen to music) radar, sonar, etc, etc, and well, like almost every real-life situation where you need to analyze or transfer streams of data/information.

Basically when you have a time-dependent signal which is too jumbled up to make sense of, you can apply the Fourier Transform (or another 'derivative' of the Fourier Series) to change the signal (usually into a frequency-dependent domain) to make it more readable or transportable. Or, in layman terms, you 'decompose' or 'split' a complicated signal into a bunch of simpler ones, to make it easier to read/interpret and take up less space when you send it over a communication channel.

So yeah, it's fucking important in the digital age.

(Excuse me for my language, but I believe it's worth emphasizing with profanity. It's not intended to be an insult or 'putting-down' or anything.)

One real-life application of the Fourier Transform that I'm intimately familiar with is in chemical analysis (I'm a chemist), in particular NMR (Nulcear Magnetic Resonance). With NMR, we *ahem* probe the nuclei (yannow, the 'cores' of atoms) of a chemical substance with radio waves. The nuclei will then sing like a bunch of canaries, allowing us to determine the identity of each nuclei, and by extension, the whole structure of the molecule.

(Nuclei of different elements [not all isotopes are NMR-active, but I'm not going to explain this as it would require knowledge of particle physics] 'sing' at different wavelengths/frequencies. Nuclei of the same element will also sing differently depending on their chemical environment-- that is, what other atoms their parent atom is attached to. A hydrogen nucleus that is attached to a big bully like say, chlorine [who likes to shake down other atoms for their 'change', i.e. electrons] will squeal at a higher 'pitch', or frequency.)

Now, there are two ways we can listen to listen to the singing. With the 'old' method, we scan the chemical substance at each frequency (in a certain range), taking down notes on how loudly the sample sings (if at all) as we change frequencies. This method is slow, and has poor resolution, as we are scanning each frequency individually.

With the newer method, thanks to our buddy Fourier, we irradiate the sample with a single high-energy radio wave pulse. (which is naturally, composed of several frequencies). We then allow the sample to 'relax', at which point all the nuclei will start singing, with decreasing 'loudness' (intensity) as time passes. This gives us a time-dependent signal, which is going to be a jumbled mess as it is a composite of all the 'songs' of each nuclei (basically similar to what you're going to see if you take a recording of several people talking and try to view the waveform in your MP3 player).

But don't worry, Fourier is here to save the day! Applying Fast Fourier Transform, we 'split' the signal into its component frequencies (or, in other words, we transform our time-dependent signal into a frequency-dependent one), allowing us to determine which frequency a molecule sings at and the intensity of each song, all in a single pass.

Since we're scanning across all (well, 'most') frequencies simultaneously, this method is remarkably faster. It also has high resolution, as we're using a single, high-energy pulse.

A specialized NMR variant used in medicine is MRI, or NMRI (Nuclear Magnetic Resonance Imaging). With this method, doctors use a specifically-tuned frequency to *ahem* probe the hydrogen nuclei in the water molecules present in your body, which the machine will then use to create a detailed image of your insides.

Another related method used by chemists is IR spectroscopy, which is similar in principle, except it uses infrared (IR) waves, or heat, instead of radio ones. Also, we target the whole molecule (or rather, the bonds present in a molecule) instead of individual nuclei.

Both methods are also popular among our CSI counterparts who work in forensics. They can, for example, tell if gasoline was used in a fire, by using these techniques.

[deleted]

NNescio said:

Stuff

You know, I wish our group had someone like you explaining Fourier back in the Uni instead of "just write this shit down and I don't care if you don't understand, I don't have time" attitude we got. It's not even hard when explained properly ._.

Makes perfect sense. Just took a while to see how she placed all the sines into a summation.

<< Math never pass 70 in any test in school
....
*try reading through the explaination*
.... (;_;)

NNescio said:

Function of Fourier

Adding that the fast fourier transform also takes part in structural analysis, specifically in soil-structure problems, it provides better result than the finite element methods.

The thing I hated most about Calculus class was that my professors sucked at explaining its practical uses. Without context, my mind felt like it couldn't be bothered to invest in learning the various formulas and rules surrounding them as it may as well have been nonsense to me. Thanks NNescio for doing what they couldn't.

The only thing that would have been better is if you had explained how the functions are specifically used since it still sounds like magic to me. 'Apply Fourier, acquire split up understandable nuclei songs hot and ready for analysis' was what I got but I'm not sure how exactly it's applied. But the explanation would have probably just rolled over my head anyway since I never learned things like Fouriers. Thanks again.

TunaMayo said:

Now i'm wondering how high Hoppo's IQ is..

Well, she's Dutch Harbor, Alaska, right? The town on said harbor, Unalaska, has about 4,400 people in it, and IIRC the "average" IQ range is from 90 to 110. So call it 100, which means: circa 440,000.

(Yes, I know IQ doesn't really work like that, it's a joke.)

Alternately, there is a branch of the University of Alaska Fairbanks in Unalaska, and one would assume they have a math department. :)

zgryphon said:

Well, she's Dutch Harbor, Alaska, right? The town on said harbor, Unalaska, has about 4,400 people in it, and IIRC the "average" IQ range is from 90 to 110. So call it 100, which means: circa 440,000.

(Yes, I know IQ doesn't really work like that, it's a joke.)

Alternately, there is a branch of the University of Alaska Fairbanks in Unalaska, and one would assume they have a math department. :)

Actually, Scott Adams of Dilbert fame once made the joke that, to find the IQ of a meeting, you start with 100, and subtract 5 for every person there. In any group with over 19 people in it, the only thing the group can accomplish is flailing around and making incomprehensible noise.

NWSiaCB said:

Actually, Scott Adams of Dilbert fame

There's something slightly too perfect about someone starting an interjection about something the Mansplainer-in-Chief said with "Actually,". :)

zgryphon said:

There's something slightly too perfect about someone starting an interjection about something the Mansplainer-in-Chief said with "Actually,". :)

"Mansplainer-in-Chief"?

NWSiaCB said:

"Mansplainer-in-Chief"?

Yeah. He's a colossal tool in the Dave Sim "women aren't really sentient beings, as such" mold. "The reality is that women are treated differently by society for exactly the same reason that children and the mentally handicapped are treated differently." That's a direct quotation.

What are we talking about, I lost track of the conversation

He's talking about a blog post by Scott Adams.

Can see the original and his follow-up here: http://blog.dilbert.com/post/102881506316/im-a-what

Saladofstones said:

What are we talking about, I lost track of the conversation

Well, it started out as a little joke about how it shouldn't be too surprising that the personification of an entire town ought to be smart enough to do some advanced math, but then it went a bit wrong. Sorry about that. It's The Internet™.